Subject: The Maxwell equations from quaternions
Sender: Doug Sweetser<email@example.com>
Organization: The World Public Access UNIX, Brookline, MA
I've been auditing a graduate level course on
electromagnetism. We've been working our way through
Jackson's "Classical Electrodynamics." You can only say you
are a real physics nerd (in the positive sense, or a Ph. D.
for that matter :-) once you've worked your way through this
impressive body of work.
The professor has remarked that after assuming Maxwell's
equations and the Lorentz force - the first equations in the
book - the rest of the text explores their mathematical
consequences, excluding the way nature dictates sources and
boundary conditions of real systems. And what an impressive
collection of mathematical consequences, with over 200 pages
on statics alone!
But there still remains the question: Why these four
equations? To be logically consistent with the rest of the
mammoth work, the equations should be a mathematical
consequence of something. Consider the following proposal:
The four Maxwell equations are generated by
one quaternion wave equation.
Quaternions were a purely mathematical invention of Hamilton.
Since quaternions are a mathematical field, it makes sense to
form a second order partial differential wave equation. With
the right choices of letters, one quaternion wave equation
will look like the four Maxwell equations. I said "letters"
because I am playing a symbolic game with quaternions, hoping
to clone a twin of physics :-)
Two players are needed, the quaternion partial differential
d/dt c + Del = d/dt c + d/dx I + d/dy J + d/dz K
and the quaternion-valued potential function:
phi + A = phi + Ax I + Ay J + Az K
Construct a second order partial differential operator
that generates the four Maxwell equations (I spent a month
playing around till I found it :-)
[(d/dt c + Del)^2 + (d/dt c - Del)^2](phi + A)/2
-----Gauss' law------ --No Monopoles--
= d^2 phi/dt^2 c^2 - del^2 phi - Div(Curl A) +
----- the B law with no name----
d^2 A/dt^2 c^2 + Grad(div A) + Curl(Curl A)
= 0 for a vacuum free of charges and currents
or = - 4 pi (rho + J/c) for sources in a vacuum
or = -4 pi (rho - div P + J/c + Curl M) macroscopic media
or = whatever sources and interactions constructed by nature.
The four Maxwell equations in the Lorentz gauge are imbedded
in this one second order quaternion partial differential
technical notes: Gaussian units are employed. The square of
an operator means that it should act twice on the potential.
To confirm that the operator math is correct requires a few
insights. There should be two second order time derivatives
and two del^2's. Only one Curl Curl operation is properly
defined. There are 6 pairwise combinations of Div, Grad and
Curl, but only 3 are well defined. The reason two operators
are needed is to eliminate the 3 terms of the form Del d/dt.
The structure of the operator that generates the Maxwell
equations is interesting in light of the recent discussion
concerning John Baez's puzzle about bosons and fermions. I
posted a response to a quaternion variant of that puzzle,
where "fermionic stuff" might be generated by
g(v, w) = (v* w + w* v)/2 .
Stretch the notion of g(v, w) a bit further to allow it to
consist of operators which can act on a quaternion-valued
v = d/dt - Del and w = d/dt + Del .
Then the Maxwell equations for sources in a vacuum are
g(v, w)(phi + A) = -4 pi (rho + J/c) .
I don't grok what this all means, but it works and I think
it's cool anyway.
The quaternions of choice
t + x I + y J + z K
E + px I + py J + pz K
phi + Ax I + Ay J + Ax K
for a new generation.
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